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A note on the multiple zeta functions and their variants at identical arguments

This paper utilizes harmonic products and complete Bell polynomials to express multiple zeta functions and their variants at identical arguments as polynomials in the Riemann zeta function, thereby characterizing their singularities and establishing a functional relation between these functions and multiple $t$ -functions.

Original authors: Pawan Singh Mehta

Published 2026-03-31

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Pawan Singh Mehta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand a massive, chaotic orchestra. In the world of mathematics, this orchestra is made up of Multiple Zeta Functions (MZFs). These are complex formulas that sum up infinite series of numbers in very specific, ordered ways.

For a long time, mathematicians knew how these instruments played individually (like the famous Riemann Zeta function), but they struggled to understand what happened when you asked the whole orchestra to play the exact same note at the same time. This is what the paper calls "identical arguments."

Here is a simple breakdown of what Pawan Singh Mehta discovered in this paper, using some everyday analogies.

1. The Problem: A Messy Orchestra

Think of the Multiple Zeta Function as a recipe for a very complicated soup.

The Ingredients: Infinite numbers (like $1, 1/2, 1/3...$ ).
The Rule: You have to add them up in a specific order (biggest to smallest).
The Confusion: When you try to calculate the value of this soup when all the "flavor settings" are turned to the same number (e.g., $s, s, s...$ ), the recipe becomes incredibly messy. It's hard to predict what the final taste will be.

2. The Magic Tool: Bell Polynomials

The author introduces a secret weapon called Bell Polynomials.

The Analogy: Imagine you have a giant, pre-made LEGO instruction manual. You know that if you want to build a specific complex shape (the Multiple Zeta Function), you don't need to invent new bricks. You just need to know exactly how many of each standard brick (the Riemann Zeta function) to use.
The Discovery: Mehta found that these Bell Polynomials are the "instruction manual." They tell us exactly how to build the complex Multiple Zeta Function using only simple, well-known Riemann Zeta functions.
Why it matters: Instead of staring at a confusing infinite sum, we can now write the answer as a neat polynomial (a math equation with powers and coefficients). It's like turning a tangled ball of yarn into a straight, organized line.

3. The "Star" and "t" Variants

The paper doesn't just look at the standard orchestra; it also looks at two "sibling" bands:

The "Star" Band ( $\zeta^\star$ ): These are like the original orchestra, but they are allowed to play the same note twice in a row (less strict rules).
The "t" Band ( $t$ and $t^\star$ ): These are like the orchestra, but they only play on "odd-numbered" notes (1, 3, 5...).

Mehta discovered a universal translator between these bands. He found a formula that shows exactly how the "t" band's music is just a mix of the "standard" band and the "star" band. It's like realizing that the sound of a violin section is just a specific combination of the flute and cello sections playing together.

4. Finding the "Broken" Notes (Singularities)

In math, some functions "break" or blow up to infinity at certain points. These are called singularities.

The Analogy: Imagine a bridge that holds up fine for most cars, but if a truck of a certain weight drives over a specific spot, the bridge collapses.
The Discovery: Because Mehta had the "LEGO instruction manual" (the Bell Polynomials), he could instantly see exactly where and how the bridge would collapse. He could predict the exact location of the "breaks" (at $s = 1, 1/2, 1/3$ , etc.) and how bad the collapse would be. This was something previous mathematicians had to work very hard to prove; Mehta's method made it obvious.

5. The "Zero" Mystery

Finally, the paper looks at what happens when you set the "flavor" to zero.

The Result: For the "odd-only" bands ( $t$ and $t^\star$ ), the soup turns out to be tasteless (the value is 0).
The Surprise: For the standard bands, the soup is actually a specific, non-zero rational number (a fraction). Mehta calculated exactly what those fractions are, using the same LEGO manual.

Summary

In short, this paper is about simplifying the complex.

Before: Multiple Zeta functions were like a black box. You put numbers in, and getting the answer out was a nightmare.
After: The author used Bell Polynomials to open the box. He showed that these complex functions are just fancy combinations of simple, well-known functions.
The Benefit: Now, mathematicians can easily predict where these functions break, what their values are at specific points (like 0 or 2), and how the different "families" of these functions relate to each other.

It's the difference between trying to guess the recipe of a cake by tasting it, versus having the exact recipe card that tells you exactly how much flour, sugar, and eggs went into it.

1. Problem Statement

The paper addresses the study of Multiple Zeta Functions (MZF) and their variants—specifically the Multiple Zeta-Star functions (MZSF), Multiple $t$ -functions (MtF), and Multiple $t$ -star functions (MtSF)—evaluated at identical arguments.

While the general theory of MZFs at distinct arguments $(s_1, \dots, s_r)$ is well-established, the behavior at the diagonal points $(s, s, \dots, s)$ presents specific structural challenges. The author aims to:

Derive explicit polynomial expressions for these functions in terms of the Riemann zeta function $\zeta(s)$ (and its variants).
Determine the coefficients of these polynomials explicitly, which were previously known only implicitly.
Analyze the singularities (poles) of these functions.
Establish functional relations connecting the standard MZFs, MZSFs, MtFs, and MtSFs.
Compute exact special values at non-positive integers (specifically at $s=0$ and negative even integers).

2. Methodology

The core methodological innovation of this paper is the application of Complete Bell Polynomials ( $Y_n$ ) and Partial Bell Polynomials ( $B_{n,k}$ ) to solve recurrence relations derived from the harmonic product (stuffle product) formulas.

Recurrence Relations: The author utilizes Newton's identities and the harmonic product formulas to establish recursive relations for the functions $f_r(s)$ (where $f$ represents $\zeta, \zeta^\star, t, t^\star$ ). For example:
$r\zeta_r(s) = \sum_{j=1}^r (-1)^{j-1} \zeta_{r-j}(s)\zeta(js)$
$r\zeta^\star_r(s) = \sum_{j=1}^r \zeta^\star_{r-j}(s)\zeta(js)$
Generating Functions & Bell Polynomials: The author transforms these recurrence relations into differential equations for generating functions. By comparing these solutions with the generating function definition of Complete Bell Polynomials:
$\exp\left(\sum_{m \ge 1} x_m \frac{t^m}{m!}\right) = \sum_{n \ge 0} Y_n(x_1, \dots, x_n) \frac{t^n}{n!}$
The author expresses the multiple zeta functions directly as evaluations of Bell polynomials with specific arguments derived from $\zeta(js)$ .
Algebraic Manipulation: Using properties of Bell polynomials (such as homogeneity and addition formulas), the author derives closed-form summation formulas and functional relations between the different variants.

3. Key Contributions and Results

A. Explicit Polynomial Representations (Theorems 1 & 2)

The paper provides the first explicit description of the coefficients for the polynomial expressions of MZFs at identical arguments.

For $\zeta_r(s)$ and $t_r(s)$ :
$\zeta_r(s) = \sum_{\substack{c_1, \dots, c_r \ge 0 \\ c_1 + 2c_2 + \dots + rc_r = r}} \frac{(-1)^{r + \sum c_i}}{1^{c_1}c_1! \cdots r^{c_r}c_r!} \zeta(s)^{c_1}\zeta(2s)^{c_2}\cdots\zeta(rs)^{c_r}$
(A similar formula exists for $t_r(s)$ ).
For $\zeta^\star_r(s)$ and $t^\star_r(s)$ :
$\zeta^\star_r(s) = \sum_{\substack{c_1, \dots, c_r \ge 0 \\ c_1 + 2c_2 + \dots + rc_r = r}} \frac{1}{1^{c_1}c_1! \cdots r^{c_r}c_r!} \zeta(s)^{c_1}\zeta(2s)^{c_2}\cdots\zeta(rs)^{c_r}$
These formulas express the functions as linear combinations of products of Riemann zeta values, with coefficients explicitly defined by combinatorial factors related to Bell polynomials.

B. Singularity Analysis (Remark 1)

Using the explicit formula (10), the author re-derives and simplifies the proof regarding the singularities of $\zeta_r(s)$ .

Result: $\zeta_r(s)$ has polar singularities only at $s = 1/k$ for $1 \le k \le r$ .
Order: The order of the pole at $s = 1/k$ is $[r/k]$ (the integer part of $r/k$ ).
Leading Coefficient: The paper provides the exact leading coefficient of the Laurent expansion at these poles, recovering previous results by Matsumoto et al. but via a more direct Bell polynomial approach.

C. Functional Relations (Theorem 3)

A novel functional relation is established connecting the standard MZFs, MZSFs, and their $t$ -variants:
$t_r(s) = \sum_{j=0}^r \frac{(-1)^{r-j}}{2^{(r-j)s}} \zeta_j(s)\zeta^\star_{r-j}(s)$
$t^\star_r(s) = \sum_{j=0}^r \frac{(-1)^{r-j}}{2^{(r-j)s}} \zeta_{r-j}(s)\zeta^\star_j(s)$
This allows the computation of $t$ -variants using only standard zeta and star-zeta functions.

D. Special Values at $s=0$ (Corollary 1)

The paper computes the values of these functions at $s=0$ , a point where standard definitions often require careful analytic continuation.

Vanishing: $t_r(0) = t^\star_r(0) = 0$ .
Non-zero Rational Values:
$\zeta_r(0) = \frac{(-1)^r}{4^r} \binom{2r}{r}$
$\zeta^\star_r(0) = -\frac{1}{r 2^{2r-1}} \binom{2r-2}{r-1}$
These results are derived using the properties of Stirling numbers of the first kind embedded within the Bell polynomial evaluation.

E. Special Values at Positive Even Integers

The paper recovers known formulas for $\zeta_r(2k)$ and $\zeta^\star_r(2k)$ in terms of Bernoulli numbers and powers of $\pi$ , demonstrating the consistency of the Bell polynomial method with classical results.

4. Significance

Unification: The work unifies the treatment of various multiple zeta variants (standard, star, $t$ , and $t$ -star) under a single framework using Bell polynomials.
Explicit Coefficients: Prior to this work, the coefficients in the polynomial expansion of $\zeta_r(s)$ were known to exist but lacked an explicit closed form. This paper provides that explicit form.
Simplification of Proofs: The Bell polynomial approach offers a streamlined, algebraic proof for the location and order of singularities, avoiding more complex analytic continuation arguments used in previous literature.
New Identities: The functional relations derived in Theorem 3 provide new tools for converting between different types of multiple zeta functions, potentially aiding in the evaluation of special values and the study of their algebraic structures.
Combinatorial Connection: The paper strengthens the link between number theory (zeta functions) and combinatorics (Bell polynomials, Stirling numbers), showing how combinatorial structures govern the analytic properties of these functions.